Properties

Label 10.10.10368641602001.1
Degree $10$
Signature $[10, 0]$
Discriminant $401^{5}$
Root discriminant $20.02$
Ramified prime $401$
Class number $1$
Class group Trivial
Galois Group $D_5$ (as 10T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 20, 2, -69, -1, 69, 2, -20, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 - 20*x^8 + 2*x^7 + 69*x^6 - x^5 - 69*x^4 + 2*x^3 + 20*x^2 - 2*x - 1)
gp: K = bnfinit(x^10 - 2*x^9 - 20*x^8 + 2*x^7 + 69*x^6 - x^5 - 69*x^4 + 2*x^3 + 20*x^2 - 2*x - 1, 1)

Normalized defining polynomial

\(x^{10} \) \(\mathstrut -\mathstrut 2 x^{9} \) \(\mathstrut -\mathstrut 20 x^{8} \) \(\mathstrut +\mathstrut 2 x^{7} \) \(\mathstrut +\mathstrut 69 x^{6} \) \(\mathstrut -\mathstrut x^{5} \) \(\mathstrut -\mathstrut 69 x^{4} \) \(\mathstrut +\mathstrut 2 x^{3} \) \(\mathstrut +\mathstrut 20 x^{2} \) \(\mathstrut -\mathstrut 2 x \) \(\mathstrut -\mathstrut 1 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $10$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[10, 0]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(10368641602001=401^{5}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $20.02$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $401$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{27} a^{8} + \frac{1}{9} a^{7} - \frac{4}{27} a^{6} + \frac{4}{9} a^{5} - \frac{1}{27} a^{4} + \frac{2}{9} a^{3} - \frac{4}{27} a^{2} - \frac{4}{9} a - \frac{8}{27}$, $\frac{1}{27} a^{9} - \frac{4}{27} a^{7} - \frac{1}{9} a^{6} - \frac{1}{27} a^{5} + \frac{1}{3} a^{4} - \frac{13}{27} a^{3} + \frac{10}{27} a - \frac{1}{9}$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

Class group and class number

Trivial Abelian group, order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $9$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{25}{27} a^{9} - \frac{32}{27} a^{8} - \frac{529}{27} a^{7} - \frac{316}{27} a^{6} + \frac{1607}{27} a^{5} + \frac{1076}{27} a^{4} - \frac{1309}{27} a^{3} - \frac{781}{27} a^{2} + \frac{220}{27} a + \frac{82}{27} \),  \( \frac{41}{27} a^{9} - \frac{70}{27} a^{8} - \frac{833}{27} a^{7} - \frac{176}{27} a^{6} + \frac{2629}{27} a^{5} + \frac{727}{27} a^{4} - \frac{2168}{27} a^{3} - \frac{566}{27} a^{2} + \frac{332}{27} a + \frac{50}{27} \),  \( \frac{7}{27} a^{9} - \frac{7}{27} a^{8} - \frac{148}{27} a^{7} - \frac{137}{27} a^{6} + \frac{377}{27} a^{5} + \frac{466}{27} a^{4} - \frac{124}{27} a^{3} - \frac{413}{27} a^{2} - \frac{80}{27} a + \frac{53}{27} \),  \( \frac{47}{27} a^{9} - \frac{77}{27} a^{8} - \frac{968}{27} a^{7} - \frac{256}{27} a^{6} + \frac{3151}{27} a^{5} + \frac{1103}{27} a^{4} - \frac{2810}{27} a^{3} - \frac{952}{27} a^{2} + \frac{521}{27} a + \frac{106}{27} \),  \( \frac{7}{27} a^{9} - \frac{8}{27} a^{8} - \frac{151}{27} a^{7} - \frac{106}{27} a^{6} + \frac{473}{27} a^{5} + \frac{359}{27} a^{4} - \frac{427}{27} a^{3} - \frac{220}{27} a^{2} + \frac{67}{27} a + \frac{7}{27} \),  \( 2 a^{9} - \frac{28}{9} a^{8} - \frac{124}{3} a^{7} - \frac{131}{9} a^{6} + \frac{392}{3} a^{5} + \frac{523}{9} a^{4} - \frac{326}{3} a^{3} - \frac{455}{9} a^{2} + \frac{49}{3} a + \frac{44}{9} \),  \( \frac{23}{27} a^{9} - \frac{5}{3} a^{8} - \frac{452}{27} a^{7} + \frac{1}{9} a^{6} + \frac{1399}{27} a^{5} + \frac{13}{3} a^{4} - \frac{983}{27} a^{3} - \frac{22}{3} a^{2} + \frac{32}{27} a + \frac{25}{9} \),  \( \frac{22}{27} a^{9} - \frac{50}{27} a^{8} - \frac{427}{27} a^{7} + \frac{161}{27} a^{6} + \frac{1484}{27} a^{5} - \frac{400}{27} a^{4} - \frac{1396}{27} a^{3} + \frac{335}{27} a^{2} + \frac{307}{27} a - \frac{98}{27} \),  \( \frac{4}{9} a^{9} - \frac{11}{27} a^{8} - \frac{29}{3} a^{7} - \frac{244}{27} a^{6} + 28 a^{5} + \frac{839}{27} a^{4} - \frac{197}{9} a^{3} - \frac{694}{27} a^{2} + \frac{17}{3} a + \frac{43}{27} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 1552.90547638 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_5$ (as 10T2):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 10
The 4 conjugacy class representatives for $D_5$
Character table for $D_5$

Intermediate fields

\(\Q(\sqrt{401}) \), 5.5.160801.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 5 sibling: 5.5.160801.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{5}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
401Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.401.2t1.1c1$1$ $ 401 $ $x^{2} - x - 100$ $C_2$ (as 2T1) $1$ $1$
2.401.5t2.2c1$2$ $ 401 $ $x^{10} - 2 x^{9} - 20 x^{8} + 2 x^{7} + 69 x^{6} - x^{5} - 69 x^{4} + 2 x^{3} + 20 x^{2} - 2 x - 1$ $D_5$ (as 10T2) $1$ $2$
2.401.5t2.2c2$2$ $ 401 $ $x^{10} - 2 x^{9} - 20 x^{8} + 2 x^{7} + 69 x^{6} - x^{5} - 69 x^{4} + 2 x^{3} + 20 x^{2} - 2 x - 1$ $D_5$ (as 10T2) $1$ $2$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.